In fig:pnn-patch-size, we show the results obtained by the three solutions (one per column) on the whole multi-rater PNN-MR subset when varying the patch size and the threshold on the scalar score \(s\) . To compare the counting errors we used, in addition to the mae, the mare that provides a percentage relating the absolute counting error to the number of objects to be counted, and the game [1]} that accounts for localization errors, as it is computed by sub-dividing the image in \(4^L\)  non-overlapping regions and computing the mae in each of these sub-regions (see also sec:back:visual-counting:metrics for further details).
As depicted in the first three rows of fig:pnn-patch-size, patch size did not significantly influence the performance of FRCNN and D-CSRNet. Thus, for these models, we suggest opting for bigger patch sizes that reduce processing overhead.
On the other hand, the S-UNet solution was more sensitive to this aspect;
due to artifacts in the overlap regions of the segmentation map, different patch sizes induced different score distributions that responded differently to score thresholding.
For S-UNet, the best performance is obtained with smaller patch sizes together with more conservative threshold values.
Note that the density-based solution D-CSRNet was the most strained by weakly-labeled training data, achieving the worst overall performance primarily due to a low recall.
Moreover, as expected, score thresholding was not effective since the density peak value employed as the score was not expected to locate the PNN precisely and correlate with its “visual quality”.
Even when counting was performed via density map integration, instead of peak localization and counting, the best counting performance that D-CSRNet achieves is an MAE of \(90.99\) .
Finally, in the last row of fig:pnn-patch-size, we show Precision-Recall (PR) curves, with the goal of better highlighting the influence of the considered threshold.
Precision was computed as the percentage of model predictions that corresponded to a ground-truth object, whereas Recall was the percentage of ground-truth objects identified by the model.
Predictions and ground-truth objects were matched using the Hungarian algorithm based on the 2D Euclidean distance between pixel coordinates; we assigned an infinite cost to pairs separated by a distance greater than 1.25 times the typical radius of objects.
As depicted in the figure, the PR curves concerning the FRCNN and the D-CSRNet are monotonic, while the S-UNet does not have this property, again due to artifacts in the segmentation maps and on merging/separating components.
